# Class 9 RD Sharma Solutions – Chapter 4 Algebraic Identities- Exercise 4.2

**Question 1: Write the following in the expanded form:**

**(i) (a + 2b + c) ^{2}**

**(ii) (2a âˆ’ 3b âˆ’ c) ^{2}**

**(iii) (âˆ’3x+y+z) ^{2}**

**(iv) (m+2nâˆ’5p) ^{2}**

**(v) (2+xâˆ’2y) ^{2}**

**(vi) (a ^{2} +b^{2} +c^{2})^{2}**

**(vii) (ab+bc+ca) ^{2}**

**(viii) (x/y+y/z+z/x) ^{2}**

**(ix) (a/bc + b/ac + c/ab) ^{2}**

**(x) (x+2y+4z) ^{2}**

**(xi) (2xâˆ’y+z) ^{2}**

**(xii) (âˆ’2x+3y+2z) ^{2}**

**Solution:**

By following the identity,

(x + y + z)^{2}= x^{2}+ y^{2}+ z^{2}+ 2xy + 2yz + 2xz

(i) (a + 2b + c)^{2}= a

^{2}+ (2b)^{2}+ c^{2}+ 2a(2b) + 2ac + 2(2b)c= a

^{2}+ 4b^{2}+ c^{2}+ 4ab + 2ac + 4bc

(ii) (2a âˆ’ 3b âˆ’ c)^{2}= [(2a) + (âˆ’3b) + (âˆ’c)]

^{2}= (2a)

^{2}+ (âˆ’3b)^{2}+ (âˆ’c)^{2}+ 2(2a)(âˆ’3b) + 2(âˆ’3b)(âˆ’c) + 2(2a)(âˆ’c)= 4a

^{2}+ 9b^{2}+ c^{2}âˆ’ 12ab + 6bc âˆ’ 4ca

(iii) (âˆ’3x+y+z)^{2}= [(âˆ’3x)

^{2}+ y^{2}+ z^{2}+ 2(âˆ’3x)y + 2yz + 2(âˆ’3x)z]= 9x

^{2}+ y^{2}+ z^{2}âˆ’ 6xy + 2yz âˆ’ 6xz

(iv) (m+2nâˆ’5p)^{2}= m

^{2}+ (2n)^{2}+ (âˆ’5p)^{2}+ 2m Ã— 2n + (2Ã—2nÃ—âˆ’5p) + 2m Ã— (âˆ’5p)= m

^{2}+ 4n^{2}+ 25p^{2}+ 4mn âˆ’ 20np âˆ’ 10pm

(v) (2+xâˆ’2y)^{2}= 2

^{2}+ x^{2}+ (âˆ’2y) 2 + 2(2)(x) + 2(x)(âˆ’2y) + 2(2)(âˆ’2y)= 4 + x

^{2}+ 4y^{2}+ 4 x âˆ’ 4xy âˆ’ 8y

(vi) (a^{2}+b^{2}+c^{2})^{2}= (a

^{2})^{2}+ (b^{2})^{2}+ (c^{2})^{2}+ 2a^{2}b^{2}+ 2b^{2}c^{2}+ 2a^{2}c^{2}= a

^{4}+ b^{4}+ c^{4}+ 2a^{2}b^{2}+ 2b^{2}c^{2}+ 2c^{2}a^{2}

(vii) (ab+bc+ca)^{2}= (ab)

^{2}+ (bc)^{2}+ (ca)^{2}+ 2(ab)(bc) + 2(bc)(ca) + 2(ab)(ca)= a

^{2}b^{2}+ b^{2}c^{2}+ c^{2}a^{2}+ 2(ac)b^{2}+ 2(ab)(c)^{2}+ 2(bc)(a)^{2}

(viii) (x/y+y/z+z/x)^{2}

(ix) (a/bc + b/ac + c/ab)^{2}

(x) (x+2y+4z)^{2}= x

^{2}+ (2y)^{2}+ (4z)^{2}+ (2x)(2y) + 2(2y)(4z) + 2x(4z)= x

^{2}+ 4y^{2}+ 16z^{2}+ 4xy + 16yz + 8xz

(xi) (2xâˆ’y+z)^{2}= (2x)

^{2}+ (âˆ’y)^{2}+ (z)^{2}+ 2(2x)(âˆ’y) + 2(âˆ’y)(z) + 2(2x)(z)= 4x

^{2}+ y^{2}+ z^{2}âˆ’ 4xy âˆ’ 2yz + 4xz

(xii) (âˆ’2x+3y+2z)^{2}= (âˆ’2x)

^{2}+ (3y)^{2}+ ( 2z)^{2}+ 2(âˆ’2x)(3y) + 2(3y)(2z) + 2(âˆ’2x)(2z)= 4x

^{2}+ 9y^{2}+ 4z^{2}âˆ’12xy + 12yz âˆ’8xz

**Question 2: Simplify**

**(i) (a + b + c) ^{2} + (a âˆ’ b + c)^{2}**

**(ii) (a + b + c) ^{2} âˆ’ (a âˆ’ b + c)^{2}**

**(iii) (a + b + c) ^{2} + (a â€“ b + c)^{2} + (a + b âˆ’ c)^{2}**

**(iv) (2x + p âˆ’ c) ^{2} âˆ’ (2x âˆ’ p + c)^{2}**

**(v) (x ^{2} + y^{2} âˆ’ z^{2})^{2} âˆ’ (x^{2} âˆ’ y^{2} + z^{2})^{2}**

**Solution:**

(i) (a + b + c)^{2}+ (a âˆ’ b + c)^{2}= (a

^{2}+ b^{2}+ c^{2}+ 2ab+2bc+2ca) + (a^{2}+ (âˆ’b)^{2}+ c^{2}âˆ’2abâˆ’2bc+2ca)= 2a

^{2}+ 2 b^{2}+ 2c^{2}+ 4ca

(ii) (a + b + c)^{2}âˆ’ (a âˆ’ b + c)^{2}= (a

^{2}+ b^{2}+ c^{2}+ 2ab+2bc+2ca) âˆ’ (a^{2}+ (âˆ’b)^{2}+ c^{2}âˆ’2abâˆ’2bc+2ca)= a2 + b2 + c2 + 2ab + 2bc + 2ca âˆ’ a2 âˆ’ b2 âˆ’ c2 + 2ab + 2bc âˆ’ 2ca

= 4ab + 4bc

(iii) (a + b + c)^{2}+ (a â€“ b + c)^{2}+ (a + b âˆ’ c)= a

^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ca + (a^{2}+ b^{2}+ (c)^{2}âˆ’ 2ab âˆ’ 2cb + 2ca) + (a^{2}+ b^{2}+ c^{2}+ 2ab âˆ’ 2bc â€“ 2ca)= 3a

^{2}+ 3b^{2}+ 3c^{2}+ 2ab âˆ’ 2bc + 2ca

(iv) (2x + p âˆ’ c)^{2}âˆ’ (2x âˆ’ p + c)^{2}= [4x

^{2}+ p^{2}+ c^{2}+ 4xp âˆ’ 2pc âˆ’ 4xc] âˆ’ [4x^{2}+ p^{2}+ c^{2}âˆ’ 4xpâˆ’ 2pc + 4xc]= 4x

^{2}+ p^{2}+ c^{2}+ 4xp âˆ’ 2pc âˆ’ 4cx âˆ’ 4x^{2}âˆ’ p^{2}âˆ’ c^{2}+ 4xp + 2pcâˆ’ 4cx= 8xp âˆ’ 8xc

= 8(xp âˆ’ xc)

(v) (x^{2}+ y^{2}âˆ’ z^{2})^{2}âˆ’ (x^{2}âˆ’ y^{2}+ z^{2})^{2}= (x

^{2}+ y^{2}+ (âˆ’z)^{2})^{2}âˆ’ (x^{2}âˆ’ y^{2}+ z^{2})^{2}= [x

^{4}+ y^{4}+ z^{4}+ 2x2y^{2}â€“ 2y2z^{2}â€“ 2x2z^{2}âˆ’ [x^{4}+ y^{4}+ z^{4}âˆ’ 2x2y^{2}âˆ’ 2y2z^{2}+ 2x2z^{2}]= 4x

^{2}y^{2}â€“ 4z^{2}x^{2}

**Question 3: If a + b + c = 0 and a**^{2} + b^{2} + c^{2} = 16, find the value of ab + bc + ca.

^{2}+ b

^{2}+ c

^{2}= 16, find the value of ab + bc + ca.

**Solution:**

Given,

a + b + c = 0 and a

^{2}+ b^{2}+ c^{2}= 16Choose a + b + c = 0

Squaring both sides,

(a + b + c)

^{2}= 0a

^{2}+ b^{2}+ c^{2}+ 2(ab + bc + ca) = 016 + 2(ab + bc + c) = 0

2(ab + bc + ca) = -16

ab + bc + ca = -16/2 = -8

or

ab + bc + ca = -8

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